Geometry of the Borel -- de Siebenthal Discrete Series

Abstract

Let $G_0$ be a connected, simply connected real simple Lie group. Suppose that $G_0$ has a compact Cartan subgroup $T_0$, so it has discrete series representations. Relative to $T_0$ there is a distinguished positive root system $Δ^+$ for which there is a unique noncompact simple root $ν$, the "Borel -- de Siebenthal system". There is a lot of fascinating geometry associated to the corresponding "Borel -- de Siebenthal discrete series" representations of $G_0$. In this paper we explore some of those geometric aspects and we work out the $K_0$--spectra of the Borel -- de Siebenthal discrete series representations. This has already been carried out in detail for the case where the associated symmetric space $G_0/K_0$ is of hermitian type, i.e. where $ν$ has coefficient 1 in the maximal root $μ$, so we assume that the group $G_0$ is not of hermitian type, in other words that $ν$ has coefficient 2 in $μ$. \medskip Several authors have studied the case where $G_0/K_0$ is a quaternionic symmetric space and the inducing holomorphic vector bundle is a line bundle. That is the case where $μ$ is orthogonal to the compact simple roots and the inducing representation is 1--dimensional.